Basic time series functions

SOE 592 – Intro to Time Series Analysis

---

Introduction

This lab introduces you to some of the basic functions in R for plotting and analyzing univariate time series data. Many of the things you learn here will be relevant when we start examining multivariate time series as well. We will begin with the creation and plotting of time series objects in R, and then moves on to decomposition, differencing, and correlation (e.g., ACF, PACF) before ending with fitting and simulation of ARMA models.

Datasets

We’ll use two publicly available environmental datasets in this lab. The main one is a time series of the atmospheric concentration of CO₂ collected at the Mauna Loa Observatory in Hawai’i (ML_CO2.csv). The second is Northern Hemisphere land and ocean temperature anomalies from NOAA. (NH_temp). You can download both of them from GitHub via the following code.

## Atmospheric CO2 measured on Mauna Loa, Hawai'i
CO2 <- read.csv("https://raw.githubusercontent.com/SOE592/website/main/lectures/day_01/data/ML_CO2.csv")

## Northern hemisphere temperature anomolies
NH_temp <- read.csv("https://raw.githubusercontent.com/SOE592/website/main/lectures/day_01/data/NH_temp.csv")

Time series plots

Time series plots are an excellent way to begin the process of understanding what sort of process might have generated the data of interest. Traditionally, time series have been plotted with the observed data on the \(y\)-axis and time on the \(x\)-axis. Sequential time points are usually connected with some form of line, but sometimes other plot forms can be a useful way of conveying important information in the time series (e.g., barplots of sea-surface temperature anomolies show nicely the contrasting El Niño and La Niña phenomena).

ts objects and plot.ts()

The CO₂ data are stored in R as a data.frame object, but we would like to transform the class to a more user-friendly format for dealing with time series. Fortunately, the ts() function will do just that, and return an object of class ts as well.

Tip: Type ?ts at the command prompt to see all of function arguments.

In addition to the data themselves, we need to provide ts() with two pieces of information about the time index for the data:

1. frequency is a bit of a misnomer because it does not really refer to the number of cycles per unit time, but rather the number of observations/samples per cycle. So, for example, if the data were collected every hour of a day then frequency = 24.

2. start specifies the starting time or date for the first data point in terms of c(unit, subunit).

So, for example, if the data were collected monthly beginning in November of 1999, then frequency = 12 and start = c(1999, 11). If the data were collected annually, then you simply specify start as a scalar (e.g., start = 1991) and omit frequency (i.e., R will set frequency = 1 by default).

The Mauna Loa time series is collected monthly and begins in March of 1958, which we can get from the data themselves, and then pass to ts().

## create a time series (ts) object from the CO2 data
co2 <- ts(data = CO2$ppm, frequency = 12,
          start = c(CO2[1, "year"], CO2[1, "month"]))

Now let’s plot the data using plot.ts(), which is designed specifically for ts objects like the one we just created above. It’s nice because we don’t need to specify any \(x\)-values as they are taken directly from the ts object.

## plot the ts
plot.ts(co2, ylab = expression(paste("CO"[2], " (ppm)")))

Time series of the atmospheric CO₂ concentration at Mauna Loa, Hawai’i measured monthly from March 1958 to present.

Examination of the plotted time series shows two obvious features that would violate any assumption of stationarity:

1. an increasing (and perhaps non-linear) trend over time, and

2. strong seasonal patterns.

(Aside: Do you know the causes of these 2 phenomena?)

Combining and plotting multiple ts objects

Before we examine the CO₂ data further, however, let’s see a quick example of how you can combine and plot multiple time series together. We’ll use the data on monthly mean temperature anomolies for the Northern Hemisphere (NH_temp).

Task: Begin by converting NH_temp to a ts object.

## convert temperature data to ts object
temp_ts <- ts(data = NH_temp$Value, frequency = 12,
              start = c(1880, 1))

Before we can plot the two time series together, however, we need to line up their time indices because the temperature data start in January of 1880, but the CO₂ data start in March of 1958. Fortunately, the ts.intersect() function makes this really easy once the data have been transformed to ts objects by trimming the data to a common time frame. Also, ts.union() works in a similar fashion, but it pads one or both series with the appropriate number of NA.

Task: Compare the results of ts.intersect() and ts.union().

## intersection (only overlapping times)
dat_int <- ts.intersect(co2, temp_ts)

## dimensions of common-time data
dim(dat_int)

## [1] 682   2

## union (all times)
dat_unn <- ts.union(co2, temp_ts)

## dimensions of all-time data
dim(dat_unn)

## [1] 1647    2

As you can see, the intersection of the two data sets is much smaller than the union. If you compare them, you will see that the first 965 rows of dat_unn contains NA in the co2 column.

It turns out that the regular plot() function in R is smart enough to recognize a ts object and use the information contained therein appropriately. Here’s how to plot the intersection of the two time series together with the y-axes on alternate sides.

## plot the ts
# plot(dat_int, main = "", yax.flip = TRUE)
plot.ts(dat_int, main = "", yax.flip = TRUE)

Time series of the atmospheric CO₂ concentration at Mauna Loa, Hawai’i (top) and the mean temperature index for the Northern Hemisphere (bottom) measured monthly from March 1958 to present.

Decomposition of time series

Plotting time series data is an important first step in analyzing their various components. Beyond that, however, we need a more formal means for identifying and removing characteristics such as a trend or seasonal variation. As discussed in lecture, the decomposition model reduces a time series into 3 components: trend, seasonal effects, and random errors. In turn, we aim to model the random errors as some form of stationary process.

Let’s begin with a simple, additive decomposition model for a time series \(x_t\)

\[\begin{equation} (\#eq:classDecomp) x_t = m_t + s_t + e_t, \end{equation}\]

where, at time \(t\), \(m_t\) is the trend, \(s_t\) is the seasonal effect, and \(e_t\) is a random error that we generally assume to have zero-mean and to be correlated over time. Thus, by estimating and subtracting both \(\{m_t\}\) and \(\{s_t\}\) from \(\{x_t\}\), we hope to have a time series of stationary residuals \(\{e_t\}\).

Estimating trends

In lecture we discussed how linear filters are a common way to estimate trends in time series. One of the most common linear filters is the moving average, which for time lags from \(-a\) to \(a\) is defined as

\[\begin{equation} (\#eq:linearFilter) \hat{m}_t = \sum_{k=-a}^{a} \left(\frac{1}{1+2a}\right) x_{t+k}. \end{equation}\]

This model works well for moving windows of odd-numbered lengths, but should be adjusted for even-numbered lengths by adding only \(\frac{1}{2}\) of the 2 most extreme lags so that the filtered value at time \(t\) lines up with the original observation at time \(t\). So, for example, in a case with monthly data such as the atmospheric CO₂ concentration where a 12-point moving average would be an obvious choice, the linear filter would be

\[\begin{equation} (\#eq:linearFilterEx) \hat{m}_t = \frac{\frac{1}{2}x_{t-6} + x_{t-5} + \dots + x_{t-1} + x_t + x_{t+1} + \dots + x_{t+5} + \frac{1}{2}x_{t+6}}{12} \end{equation}\]

It is important to note here that our time series of the estimated trend \(\{\hat{m}_t\}\) is actually shorter than the observed time series by \(2a\) units.

Conveniently, R has the built-in function filter() in the stats package for estimating moving-average (and other) linear filters. In addition to specifying the time series to be filtered, we need to pass in the filter weights (and 2 other arguments we won’t worry about here–type ?filter to get more information). The easiest way to create the filter is with the rep() function:

## weights for moving average
fltr <- c(1 / 2, rep(1, times = 11), 1 / 2) / 12

Now let’s get our estimate of the trend \(\{\hat{m}\}\) with filter()} and plot it:

## estimate of trend
co2_trend <- stats::filter(co2, filter = fltr, method = "convo", sides = 2)

## plot the trend
plot.ts(co2_trend, ylab = "Trend", cex = 1)

The trend is a more-or-less smoothly increasing function over time, the average slope of which does indeed appear to be increasing over time as well (Figure @ref(fig:tslab-plotTrendTSb)).

(ref:tslab-plotTrendTSb) Time series of the estimated trend \(\{\hat{m}_t\}\) for the atmospheric CO₂ concentration at Mauna Loa, Hawai’i.

(ref:tslab-plotTrendTSb)

Estimating seasonal effects

Once we have an estimate of the trend for time \(t\) (\(\hat{m}_t\)) we can easily obtain an estimate of the seasonal effect at time \(t\) (\(\hat{s}_t\)) by subtraction

\[\begin{equation} (\#eq:seasEst) \hat{s}_t = x_t - \hat{m}_t, \end{equation}\]

which is really easy to do in R:

## seasonal effect over time
co2_seas <- co2 - co2_trend

This estimate of the seasonal effect for each time \(t\) also contains the random error \(e_t\), however, which can be seen by plotting the time series and careful comparison of Equations @ref(eq:classDecomp) and @ref(eq:seasEst).

## plot the monthly seasonal effects
plot.ts(co2_seas, ylab = "Seasonal effect", xlab = "Month", cex = 1)

(ref:tslab-plotSeasTSb) Time series of seasonal effects plus random errors for the atmospheric CO₂ concentration at Mauna Loa, Hawai’i, measured monthly from March 1958 to present.

(ref:tslab-plotSeasTSb)

We can obtain the overall seasonal effect by averaging the estimates of \(\{\hat{s}_t\}\) for each month and repeating this sequence over all years.

## length of ts
ll <- length(co2_seas)

## frequency (ie, 12)
ff <- frequency(co2_seas)

## number of periods (years); %/% is integer division
periods <- ll %/% ff

## index of cumulative month
index <- seq(1, ll, by = ff) - 1

## get mean by month
mm <- numeric(ff)
for (i in 1:ff) {
  mm[i] <- mean(co2_seas[index + i], na.rm = TRUE)
}

## subtract mean to make overall mean = 0
mm <- mm - mean(mm)

Before we create the entire time series of seasonal effects, let’s plot them for each month to see what is happening within a year:

## plot the monthly seasonal effects
plot.ts(mm, ylab = "Seasonal effect", xlab = "Month", cex = 1)

It looks like, on average, that the CO₂ concentration is highest in spring (March) and lowest in summer (August) (Figure @ref(fig:tslab-plotSeasMean)). (Aside: Do you know why this is?)

(ref:tslab-plotSeasMean) Estimated monthly seasonal effects for the atmospheric CO₂ concentration at Mauna Loa, Hawai’i.

(ref:tslab-plotSeasMean)

Finally, let’s create the entire time series of seasonal effects \(\{\hat{s}_t\}\):

## create ts object for season
co2_seas_ts <- ts(rep(mm, periods + 1)[seq(ll)],
  start = start(co2_seas),
  frequency = ff
)

Completing the model

The last step in completing our full decomposition model is obtaining the random errors \(\{\hat{e}_t\}\), which we can get via simple subtraction

\[\begin{equation} (\#eq:errorEst) \hat{e}_t = x_t - \hat{m}_t - \hat{s}_t. \end{equation}\]

Again, this is really easy in R:

## random errors over time
co2_err <- co2 - co2_trend - co2_seas_ts

Now that we have all 3 of our model components, let’s plot them together with the observed data \(\{x_t\}\). The results are shown in Figure @ref(fig:tslab-plotTrSeas).

## plot the obs ts, trend & seasonal effect
plot(cbind(co2, co2_trend, co2_seas_ts, co2_err), main = "", yax.flip = TRUE)

(ref:tslab-plotTrSeas) Time series of the observed atmospheric CO₂ concentration at Mauna Loa, Hawai’i (top) along with the estimated trend, seasonal effects, and random errors.

(ref:tslab-plotTrSeas)

Using decompose() for decomposition

Now that we have seen how to estimate and plot the various components of a classical decomposition model in a piecewise manner, let’s see how to do this in one step in R with the function decompose(), which accepts a ts object as input and returns an object of class decomposed.ts.

## decomposition of CO2 data
co2_decomp <- decompose(co2)

co2_decomp is a list with the following elements, which should be familiar by now:

• x: the observed time series \(\{x_t\}\)
• seasonal: time series of estimated seasonal component \(\{\hat{s}_t\}\)
• figure: mean seasonal effect (length(figure) == frequency(x))
• trend: time series of estimated trend \(\{\hat{m}_t\}\)
• random: time series of random errors \(\{\hat{e}_t\}\)
• type: type of error ("additive" or "multiplicative")

We can easily make plots of the output and compare them to those in Figure @ref(fig:tslab-plotTrSeas):

## plot the obs ts, trend & seasonal effect
plot(co2_decomp, yax.flip = TRUE)

(ref:tslab-plotDecompB) Time series of the observed atmospheric CO₂ concentration at Mauna Loa, Hawai’i (top) along with the estimated trend, seasonal effects, and random errors obtained with the function decompose().

(ref:tslab-plotDecompB)

The results obtained with decompose() (Figure @ref(fig:tslab-plotDecompB)) are identical to those we estimated previously.

Another nice feature of the decompose() function is that it can be used for decomposition models with multiplicative (i.e., non-additive) errors (e.g., if the original time series had a seasonal amplitude that increased with time). To do, so pass in the argument type = "multiplicative", which is set to type = "additive" by default.

Differencing to remove a trend or seasonal effects

An alternative to decomposition for removing trends is differencing. We saw in lecture how the difference operator works and how it can be used to remove linear and nonlinear trends as well as various seasonal features that might be evident in the data. As a reminder, we define the difference operator as

\[\begin{equation} (\#eq:diffDefnA) \nabla x_t = x_t - x_{t-1}, \end{equation}\]

and, more generally, for order \(d\)

\[\begin{equation} (\#eq:diffDefnB) \nabla^d x_t = (1-\BB)^d x_t, \end{equation}\] where B is the backshift operator (i.e., \(\BB^k x_t = x_{t-k}\) for \(k \geq 1\)).

So, for example, a random walk is one of the most simple and widely used time series models, but it is not stationary. We can write a random walk model as

\[\begin{equation} (\#eq:defnRW) x_t = x_{t-1} + w_t, \text{ with } w_t \sim \text{N}(0,q). \end{equation}\]

Applying the difference operator to Equation @ref(eq:defnRW) will yield a time series of Gaussian white noise errors \(\{w_t\}\):

\[\begin{equation} (\#eq:diffRW) \begin{aligned} \nabla (x_t &= x_{t-1} + w_t) \\ x_t - x_{t-1} &= x_{t-1} - x_{t-1} + w_t \\ x_t - x_{t-1} &= w_t \end{aligned} \end{equation}\]

Using the diff() function

In R we can use the diff() function for differencing a time series, which requires 3 arguments: x (the data), lag (the lag at which to difference), and differences (the order of differencing; \(d\) in Equation @ref(eq:diffDefnB)). For example, first-differencing a time series will remove a linear trend (i.e., differences = 1); twice-differencing will remove a quadratic trend (i.e., differences = 2). In addition, first-differencing a time series at a lag equal to the period will remove a seasonal trend (e.g., set lag = 12 for monthly data).

Let’s use diff() to remove the trend and seasonal signal from the CO₂ time series, beginning with the trend. Close inspection of Figure @ref(fig:tslab-plotdata1) would suggest that there is a nonlinear increase in CO₂ concentration over time, so we’ll set differences = 2):

## twice-difference the CO2 data
co2_d2 <- diff(co2, differences = 2)

## plot the differenced data
plot(co2_d2, ylab = expression(paste(nabla^2, "CO"[2])))

(ref:tslab-plotCO2diff2) Time series of the twice-differenced atmospheric CO₂ concentration at Mauna Loa, Hawai’i.

(ref:tslab-plotCO2diff2)

We were apparently successful in removing the trend, but the seasonal effect still appears obvious (Figure @ref(fig:tslab-plotCO2diff2)). Therefore, let’s go ahead and difference that series at lag-12 because our data were collected monthly.

## difference the differenced CO2 data
co2_d2d12 <- diff(co2_d2, lag = 12)

## plot the newly differenced data
plot(co2_d2d12,
     ylab = expression(paste(nabla, "(", nabla^2, "CO"[2], ")")))

(ref:tslab-plotCO2diff12) Time series of the lag-12 difference of the twice-differenced atmospheric CO₂ concentration at Mauna Loa, Hawai’i.

(ref:tslab-plotCO2diff12)

Now we have a time series that appears to be random errors without any obvious trend or seasonal components (Figure @ref(fig:tslab-plotCO2diff12)).

Correlation within and among time series

The concepts of covariance and correlation are very important in time series analysis. In particular, we can examine the correlation structure of the original data or random errors from a decomposition model to help us identify possible form(s) of (non)stationary model(s) for the stochastic process.

Autocorrelation function (ACF)

Autocorrelation is the correlation of a variable with itself at differing time lags. Recall from lecture that we defined the sample autocovariance function (ACVF), \(c_k\), for some lag \(k\) as

\[\begin{equation} (\#eq:ACVF) c_k = \frac{1}{n}\sum_{t=1}^{n-k} \left(x_t-\bar{x}\right) \left(x_{t+k}-\bar{x}\right) \end{equation}\]

Note that the sample autocovariance of \(\{x_t\}\) at lag 0, \(c_0\), equals the sample variance of \(\{x_t\}\) calculated with a denominator of \(n\). The sample autocorrelation function (ACF) is defined as

\[\begin{equation} (\#eq:ACF) r_k = \frac{c_k}{c_0} = \text{Cor}(x_t,x_{t+k}) \end{equation}\]

Recall also that an approximate 95% confidence interval on the ACF can be estimated by

\[\begin{equation} (\#eq:ACF95CI) -\frac{1}{n} \pm \frac{2}{\sqrt{n}} \end{equation}\]

where \(n\) is the number of data points used in the calculation of the ACF.

It is important to remember two things here. First, although the confidence interval is commonly plotted and interpreted as a horizontal line over all time lags, the interval itself actually grows as the lag increases because the number of data points \(n\) used to estimate the correlation decreases by 1 for every integer increase in lag. Second, care must be exercised when interpreting the “significance” of the correlation at various lags because we should expect, a priori, that approximately 1 out of every 20 correlations will be significant based on chance alone.

We can use the acf() function in R to compute the sample ACF (note that adding the option type = "covariance" will return the sample auto-covariance (ACVF) instead of the ACF–type ?acf for details). Calling the function by itself will will automatically produce a correlogram (i.e., a plot of the autocorrelation versus time lag). The argument lag.max allows you to set the number of positive and negative lags. Let’s try it for the CO₂ data.

## correlogram of the CO2 data
acf(co2, lag.max = 36)

(ref:tslab-plotACFb) Correlogram of the observed atmospheric CO₂ concentration at Mauna Loa, Hawai’i obtained with the function acf().

(ref:tslab-plotACFb)

There are 4 things about Figure @ref(fig:tslab-plotACFb) that are noteworthy:

1. the ACF at lag 0, \(r_0\), equals 1 by default (i.e., the correlation of a time series with itself)–it’s plotted as a reference point;
2. the \(x\)-axis has decimal values for lags, which is caused by R using the year index as the lag rather than the month;
3. the horizontal blue lines are the approximate 95% CI’s; and
4. there is very high autocorrelation even out to lags of 36 months.

As an alternative to the default plots for acf objects, let’s define a new plot function for acf objects with some better features:

## better ACF plot
plot_acf <- function(ACFobj) {
  rr <- ACFobj$acf[-1]
  kk <- length(rr)
  nn <- ACFobj$n.used
  plot(seq(kk), rr,
    type = "h", lwd = 2, yaxs = "i", xaxs = "i",
    ylim = c(floor(min(rr)), 1), xlim = c(0, kk + 1),
    xlab = "Lag", ylab = "Correlation", las = 1
  )
  abline(h = -1 / nn + c(-2, 2) / sqrt(nn), lty = "dashed", col = "blue")
  abline(h = 0)
}

Now we can assign the result of acf() to a variable and then use the information contained therein to plot the correlogram with our new plot function.

## acf of the CO2 data
co2_acf <- acf(co2, lag.max = 36)

## correlogram of the CO2 data
plot_acf(co2_acf)

(ref:tslab-plotbetterACF) Correlogram of the observed atmospheric CO₂ concentration at Mauna Loa, Hawai’i obtained with the function plot_acf().

(ref:tslab-plotbetterACF)

Notice that all of the relevant information is still there (Figure @ref(fig:tslab-plotbetterACF)), but now \(r_0=1\) is not plotted at lag-0 and the lags on the \(x\)-axis are displayed correctly as integers.

Before we move on to the PACF, let’s look at the ACF for some deterministic time series, which will help you identify interesting properties (e.g., trends, seasonal effects) in a stochastic time series, and account for them in time series models–an important topic in this course. First, let’s look at a straight line.

## length of ts
nn <- 100

## create straight line
tt <- seq(nn)

## set up plot area
par(mfrow = c(1, 2))

## plot line
plot.ts(tt, ylab = expression(italic(x[t])))

## get ACF
line.acf <- acf(tt, plot = FALSE)

## plot ACF
plot_acf(line.acf)

(ref:tslab-plotLinearACF) Time series plot of a straight line (left) and the correlogram of its ACF (right).

(ref:tslab-plotLinearACF)

The correlogram for a straight line is itself a linearly decreasing function over time (Figure @ref(fig:tslab-plotLinearACF)).

Now let’s examine the ACF for a sine wave and see what sort of pattern arises.

## create sine wave
tt <- sin(2 * pi * seq(nn) / 12)

## set up plot area
par(mfrow = c(1, 2))

## plot line
plot.ts(tt, ylab = expression(italic(x[t])))

## get ACF
sine_acf <- acf(tt, plot = FALSE)

## plot ACF
plot_acf(sine_acf)

(ref:tslab-plotSineACF) Time series plot of a discrete sine wave (left) and the correlogram of its ACF (right).

(ref:tslab-plotSineACF)

Perhaps not surprisingly, the correlogram for a sine wave is itself a sine wave whose amplitude decreases linearly over time (Figure @ref(fig:tslab-plotSineACF)).

Now let’s examine the ACF for a sine wave with a linear downward trend and see what sort of patterns arise.

## create sine wave with trend
tt <- sin(2 * pi * seq(nn) / 12) - seq(nn) / 50

## set up plot area
par(mfrow = c(1, 2))

## plot line
plot.ts(tt, ylab = expression(italic(x[t])))

## get ACF
sili_acf <- acf(tt, plot = FALSE)

## plot ACF
plot_acf(sili_acf)

(ref:tslab-plotSiLiACF) Time series plot of a discrete sine wave (left) and the correlogram of its ACF (right).

(ref:tslab-plotSiLiACF)

The correlogram for a sine wave with a trend is itself a nonsymmetrical sine wave whose amplitude and center decrease over time (Figure @ref(fig:tslab-plotSiLiACF)).

As we have seen, the ACF is a powerful tool in time series analysis for identifying important features in the data. As we will see later, the ACF is also an important diagnostic tool for helping to select the proper order of \(p\) and \(q\) in ARMA(\(p\),\(q\)) models.

Partial autocorrelation function (PACF)

The partial autocorrelation function (PACF) measures the linear correlation of a series \(\{x_t\}\) and a lagged version of itself \(\{x_{t+k}\}\) with the linear dependence of \(\{x_{t-1},x_{t-2},\dots,x_{t-(k-1)}\}\) removed. Recall from lecture that we define the PACF as

\[\begin{equation} (\#eq:PACFdefn) f_k = \begin{cases} \text{Cor}(x_1,x_0)=r_1 & \text{if } k = 1;\\ \text{Cor}(x_k-x_k^{k-1},x_0-x_0^{k-1}) & \text{if } k \geq 2; \end{cases} \end{equation}\]

with

It’s easy to compute the PACF for a variable in R using the pacf() function, which will automatically plot a correlogram when called by itself (similar to acf()). Let’s look at the PACF for the CO₂ data.

## PACF of the CO2 data
pacf(co2, lag.max = 36)

The default plot for PACF is a bit better than for ACF, but here is another plotting function that might be useful.

## better PACF plot
plot_pacf <- function(PACFobj) {
  rr <- PACFobj$acf
  kk <- length(rr)
  nn <- PACFobj$n.used
  plot(seq(kk), rr,
    type = "h", lwd = 2, yaxs = "i", xaxs = "i",
    ylim = c(floor(min(rr)), 1), xlim = c(0, kk + 1),
    xlab = "Lag", ylab = "PACF", las = 1
  )
  abline(h = -1 / nn + c(-2, 2) / sqrt(nn), lty = "dashed", col = "blue")
  abline(h = 0)
}

(ref:tslab-plotPACFb) Correlogram of the PACF for the observed atmospheric CO₂ concentration at Mauna Loa, Hawai’i obtained with the function pacf().

(ref:tslab-plotPACFb)

Notice in Figure @ref(fig:tslab-plotPACFb) that the partial autocorrelation at lag-1 is very high (it equals the ACF at lag-1), but the other values at lags > 1 are relatively small, unlike what we saw for the ACF. We will discuss this in more detail later on in this lab.

Notice also that the PACF plot again has real-valued indices for the time lag, but it does not include any value for lag-0 because it is impossible to remove any intermediate autocorrelation between \(t\) and \(t-k\) when \(k=0\), and therefore the PACF does not exist at lag-0. If you would like, you can use the plot_acf() function we defined above to plot the PACF estimates because acf() and pacf() produce identical list structures (results not shown here).

## PACF of the CO2 data
co2_pacf <- pacf(co2)

## correlogram of the CO2 data
plot_acf(co2_pacf)

As with the ACF, we will see later on how the PACF can also be used to help identify the appropriate order of \(p\) and \(q\) in ARMA(\(p\),\(q\)) models.

Cross-correlation function (CCF)

Often we are interested in looking for relationships between 2 different time series. There are many ways to do this, but a simple method is via examination of their cross-covariance and cross-correlation.

We begin by defining the sample cross-covariance function (CCVF) in a manner similar to the ACVF, in that

\[\begin{equation} (\#eq:CCVF) g_k^{xy} = \frac{1}{n}\sum_{t=1}^{n-k} \left(y_t-\bar{y}\right) \left(x_{t+k}-\bar{x}\right), \end{equation}\]

but now we are estimating the correlation between a variable \(y\) and a different time-shifted variable \(x_{t+k}\). The sample cross-correlation function (CCF) is then defined analogously to the ACF, such that

\[\begin{equation} (\#eq:CCF) r_k^{xy} = \frac{g_k^{xy}}{\sqrt{\text{SD}_x\text{SD}_y}}; \end{equation}\]

SD\(_x\) and SD\(_y\) are the sample standard deviations of \(\{x_t\}\) and \(\{y_t\}\), respectively. It is important to re-iterate here that \(r_k^{xy} \neq r_{-k}^{xy}\), but \(r_k^{xy} = r_{-k}^{yx}\). Therefore, it is very important to pay particular attention to which variable you call \(y\) (i.e., the “response”) and which you call \(x\) (i.e., the “predictor”).

As with the ACF, an approximate 95% confidence interval on the CCF can be estimated by

\[\begin{equation} (\#eq:CCF95CI) -\frac{1}{n} \pm \frac{2}{\sqrt{n}} \end{equation}\]

where \(n\) is the number of data points used in the calculation of the CCF, and the same assumptions apply to its interpretation.

Computing the CCF in R is easy with the function ccf() and it works just like acf(). In fact, ccf() is just a “wrapper” function that calls acf(). As an example, let’s examine the CCF between sunspot activity and number of lynx trapped in Canada as in the classic paper by Moran.

To begin, let’s get the data, which are conveniently included in the datasets package included as part of the base installation of R. Before calculating the CCF, however, we need to find the matching years of data. Again, we’ll use the ts.intersect() function.

## get the matching years of sunspot data
suns <- ts.intersect(lynx, sunspot.year)[, "sunspot.year"]

## get the matching lynx data
lynx <- ts.intersect(lynx, sunspot.year)[, "lynx"]

Here are plots of the time series.

## plot time series
plot(cbind(suns, lynx), yax.flip = TRUE)

(ref:tslab-plotSunsLynx) Time series of sunspot activity (top) and lynx trappings in Canada (bottom) from 1821-1934.

(ref:tslab-plotSunsLynx)

It is important to remember which of the 2 variables you call \(y\) and \(x\) when calling ccf(x, y, ...). In this case, it seems most relevant to treat lynx as the \(y\) and sunspots as the \(x\), in which case we are mostly interested in the CCF at negative lags (i.e., when sunspot activity predates inferred lynx abundance). Furthermore, we’ll use log-transformed lynx trappings.

## CCF of sunspots and lynx
ccf(suns, log(lynx), ylab = "Cross-correlation")

(ref:tslab-plotCCFb) CCF for annual sunspot activity and the log of the number of lynx trappings in Canada from 1821-1934.

(ref:tslab-plotCCFb)

From Figures @ref(fig:tslab-plotSunsLynx) and @ref(fig:tslab-plotCCFb) it looks like lynx numbers are relatively low 3-5 years after high sunspot activity (i.e., significant correlation at lags of -3 to -5).